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(1)On the Upper Bound of the Number of Positive Roots of a Random Algebraic Equation By Takashi UNO and Hiroshi NEGISHI* Abstract: We estimate the upper bound of the number of positive roots of a random algebraic equation whose random coefficients are dependent Gaussian random variables.. 1. Introduction. Let E,(z,tu) = ao(co) + ai(co)z + ･･･ + a.(tu)zn. be a random algebraic polynomial of degree n. Let N.(B,co) denote the number of zeros of I7h(z,co) in a Borel set B C D, where D C Z (the complex plane) is in the domain of Ih(z,co). During the past 30-40 years, most of the studies on random algebraic polynomials have concerned with the estimation of Nn (R,co).. An interesting study on N.(R,co) is that of Evans (1965). Evans proved the following theorem for random algebraic polynomials with independent standard Gaussian random coefficients which are real valued variables.. '. Theorem A. 71Pzere exist an integer no and a set S2o C S;2 with. A. . pa(90)Slog no-log log log no such that for each n > no and all co E S92 - S2o,. Nh(R,oo) s cr(log log n)2 log n, where cr and A are constants.. Another recent study is that of Sambandham (1978). Sambandham showed that the upper bound in the above theorem remains unchanged if the real valued random coefficients a.(oo) v = O, 1, ･･･, n are normally distributed with. mean zero and joint density function. IMIg (2.)-te'iF'i ,.p (-g.tM.), * Department of Mathematics, Facult.y of Education, Yokohama National University..

(2) 2 T. UNo and H. NEGisHi where M-i is the moment matrix with pij･ = 1, (i = D, pij･ = p, O < p < 1, (i l J'), i, i = O, 1, ･･･, n and a' is the transpose of the column vector a.. In this paper we prove that the upper bound of the number of positive roots in the above theorem remains unchanged if the random coefficients a.(co)v = O, 1, ･･･, n are normally distributed with mean zero and joint density. function. IMIS (2z)-t!'i!"' exp (-ga'Ma), ,- }. where M-i is the moment matrix with ,/. pij-(bi,-,i ,1･i;i･l where O < 6 s{ plimfl s 1, (i,i = O, 1, ･･･, n) and a' is the transpose of the column vector a. If {a.(tu); v }ir O} is stationary, then the above condition of the correlation function is satisfied.. 2. A theorem on the upper bound of the number of positive zeros of. random polynomial Consider the family of polynomials. E,n (z,co) =2 a.(co)zV (2.1) v=O where a.(co)v == O, 1, ･･･, n are dependent random variables assuming real variables only and joint density function. ts. IMIS (2z)-i!'E7!'i exp (-ga'Ma), '. where M-i = [pij] is the moment matrix with. pij=(bi,-,[ ,1･i,-B where for some 6, O < 6 f{ pli-if s 1, (i, 1' = O, 1, ･･･, n), and a' is the transpose of the column vector a. We shall prove the following theorem. Theorem 2.1. 7-here exist an integer no and a set S2o C S2 with. A. pt(9o) s{ log no - log log log no such that for each n > no and all tu E S2 - S2o,. (2.2).

(3) On the Upper Bound of the Number of Positive Roots. 3. Nn(R",co) s{ ev(log log n)2 log n,. (2.3). where N.(R',oo) is the number ofpositive roots of I7L,(z,co), dcf7ned by. a o,. and ev and A are absolute constants. Before proving the above theorem, we remark that although x E [O,oo], it is sufficient to restrict our attention to the number of zeros of I7V,(x,co) in [O,1]. If. I},(x,co) has a zero [1,oo] then xn E,(y,co), where y = x-i, has a zero in [O,1]. Therefore, it is sufficient to consider the number of zeros in the interval [O,1].. Our technique of the proof is analogous to those of Evans and Sambandham. Proof: step 1. We define the circles Co, C., C. and Ci as follows.. 1. Co; with center at z = O and of radius r = i,. C.;withcenteratz=. 31. log log no. log log no. z- 2no ,andOfradiUSr==i- 2no ,. Cm; with center at z == x. = 1 - 2mM and of radius r. = 1 - 2XM = 2-(m+i). '. for m = mo, mi, ･･･, M, where. mo = [iog no - iog lgg 5og no + iog 3] - i. and log n - log log log n- 1< M < log n - log log log n. log 2. log 2. and. log log n. Ci;withcenteratz=1andofradius . n. Then the circle Co, C., C. (m = mo, ･･･, M) and Ci cover the inteval [O,1].. Let ri(i = O, c, mo, ･･･, M, 1) be the circle concentric with Ci(i = O, c, mo, ･･･,. M, 1) and of its twice radius respectively. Then ri are interior to the circle. lzl == 1 + 2 log log n.. n Now let N(lz - zol < r) be the number of zeros of a regular function ip(z) in the circle with center zo and of radius r. By Jensen's theorem the following. inequality can be proved. iog (E}'IEPizitoE,R)Ilip(Z)l). N(lz - zol < r) g. log (Rlr). (2.4).

(4) 4 T. UNo and H. NEGisHi where R(> r) denotes the radius of a concentric circle. IS. nte. ?his step, we estimate the upper bound of the number of zeros of Ez(z, co) in. the circle Co. From the assumption, each a.(co) has marginal density function. (27t)-S exp (-{2L). If maxos.s.la.(co)l > (n + 1) then la.(co)1 > (n + 1) for at least one value. .t. v f{ n, so that, P({(iD; ,{.}.a-.x. Iav(tu)1 > (n + i)}) -`n .E., P({(o; la.(co)1 > (n + i)}). v. = (n + i) (;)S JIIi.i exp (-{2L) du. . (-;})g ,.p (-(n +2 i)2). (2.s) Since IE,(z,co)1 s (n + 1) lzln maxos.s. Ia.(oo)l, in the circle. 2 log log n. lzl=1+ , n. by (2.5), we get IE,(z,co)l :E{ (i + 2 iOgniOg n)" (n + i) ol.p.a-.x. Ia.(co)1. <(n + 1)2 exp(2 1og log n) (2.6). -". outside a set of measure at most. (;;)S ,.p (-(n +2 1)2).. Also, IE,(O,tu)l = lao(tu)1 and. p({co; lao(co)1 < (n + i)-2}) = (-;)S L"")-2 e.p (-!s2L) d.. < (;)S (. +1 1)2'. Hence,. IE,(O,(o)1 - lao((o)1 }i (n + 1)m2 (2.7) outside a set of measure at most. d.

(5) On the Upper Bound of the Number of Positive Roots 5 i +' i)2' (;)i (. If Nb denotes the number of zeros of ,P},(z,co) in the circle Co, then by (2.4), (2.6) and (2.7) we have. 4 log(n + 1) + 2 log log n. Ab <. log 2. outside a set of measure at most. (i)i exp (-(" +2 i)2) + (;)S (. +i i)2･. Then for all n > no, we get. 4 log(n + 1) + 2 log log n. Nb <. log 2. outside a set of measure at most. .=#,.i((-;)S exp (-(n +2 i)2) + (;)S (. +i i)2] < i:ii. where C is an absolute constant. step 3. In this step, we obtain an upper estimate of the number of zeros of ,Fh(z,co) in the circle C.. First we get. p ((co; .￡.oa.(co) (2 - iOg2i.Oog nO)" < (n +i i)2]). = (-i>)ii2 i;Ln'i)-2 exp (-2Ui}) du. <(i)ii2±(. +1 1)2 , (2'8) where di = ,2."., (2 - iog2i.o,g no)2' + 2 te.f (2 - iog2i.o,g no)i'f pi,-ii.. A simple calculation shows that oZ > ,2.n.o (2 - iOg2inoog no)2'.

(6) T. UNo and H. NEGisHi. 6. i - exp (-2(n + i) (2 + iOg2inOog no)). > 1-(2-IOg21.0,gno)2 (29) If N. denotes the number of zeros of E,(z,tu) in the circle C., then by (2.4), (2.6), (2.8) and (2.9) we get. 4 log(n + 1) + 2 log log n log 2. Nc <. '. outside a set of measure at most. (;)g ,.p (-gLs-Dl) + (-;-)g (. +11)2.h Then for all n > no, we get. 4 log(n + 1) + 2 log log n log 2. Nc <. outside a set of measure at most .=S,,., (-l}Y (exp (-(" +2 i)2) + (. +ii),.h] < .i,, (i !ogagogg.n,o)-,)5. step 4 To obtain an upper estimate of the number of zeros of E,(z,co) in the circle. C.(m = mo, ･･･, M), we need the following two lemmas. Lemma 2.2. Let 9o be an arbitrary measurable subset of 9. 71hen, for complex. numbers g., we have J., iog .￡., av(co)gv dco < pt(9o) iog o + pa(go) iog iog (pa(Cg,)),. where o2 - ]Il￡ lgil2 oo + 2 2oo lgil i&l pli-fl･. i--O i<J' Proof: Let g. .= bv + ic., where b. and c. E R. Also let. rl. F== {tu; 2 a.(co)g. }i Ao}, v==O ･. G= {co; 2 n av(co)b. }i Abl2'/2} v==O. and. ,.

(7) On the Upper Bound of the Number of Positive Roots. 7. n H= {co; 2 av(co)c. }i Abl2'i2}. v==O Now pt(G) = (-;;)ii2 ii J& exp ('2Uo2;:2) du. < A;i/2 exp (--{li2L) where. nn. o};2 - 2 b?･ +22 bi bf Pli-fl･. i--O i<J' Similarly,. pa(H) < Aiy2 exp (--4i2L). Since FC G U H,. pt(F) s pt(G) + pa(H) < Ax4i/2 exp (- A42 ). Following Evans, we get the proof of the lemma. Now. s2o - (s2o xx F) + (s2o n F).. Therefore ･. fg,iOg .￡.oav((D)gv d(D =Jg,xFiog .￡.oa.((D)g. d(D. n + Jg,.Fiog .Z.oav((D)g. dco = Ii + h, say. Clearly. Ii S pt(9o X. F) log Ao. Put. n L= {tu;ios 2 a.(co)g. < (i + 1)o} v=O whereiE IV and io = A, so that F= U:=i, 4･. Then,. (2.10).

(8) 8 T. UNo and H. NEGisHi i2 = Jg,.Fiog .li.lon av((D)gv d(D = i2=ooi, Jg,nE. iog .II.lio av(co)g. dco. oo < 2 log((i + 1)a)pa(9o n E･). i-- io '. Now by (2.10) ,lrl),, iog(i + i)pa(je) < ,lfE),, iog(i + i) (i.4.,) e.p (-{i2L). < c(iog io) exp (--li?I). = C(log A) exp (--I>2L). Therefore. i2 < pt(s2o n F) log o + c(log A) exp (--4il2). Choose A to satisfy. c exp (--{i2L) = pt(go), so that. log A =: log log (pa(Cg',)), where C' is a constant. Then I, g pa(go xx F) log o + pt(S2o × F) log log (pt(Cs2',)). and I2 < pa(s2o n F) log o + pa(S2o) log log (pa(Cs;2',)).. Hence. vlo (pt(Cg',). J., iog 2 av(co)gv dco < pt(9o) iog o+ pa(go) iog iog as required.. )･.

(9) On the Upper Bound of the Number of Positive Roots 9 Lemma 2.3. lf g., v E No are real, and if. G={oo; 2 oo a.((D)g. s{ Eo}, v=O then. pa(G) < Ee, where. oo gioog) pli-fl, u2 = 2 gl +22 i=0 i<f. nng) pii-J'i ol == iE gi +22 gi i--o i<f and e = (21Mi/2 (bl,in); and if 9o is any measurable subset of 9 such that 9o n G = ip, then f., iOg .￡., av(co)gv doo > pt(9o) iog a - cept(go) iog (pt(5,)).. Proof: Since the first assertion is very trivial, we omit the proof of that. Here. following Evans we show the latter part of the above lemma. Put. n Ei = {co; Ai+io< 2 a.((o)g. :E{ Aio} v==O where {Ai} is a decreasing sequence with Ao = co and Ai, = E > O. If S2o A G = ip,. then. n io-1. J., iog .E., av((D)gv d(D > ,2=, iog(Ai+i cT)pa(s:2o A E,). io-1. }r 2 log Ai+i pa(S2o n Ei) + pa(S2o) log o.. A,i21 We take Ai = 1, A2 = AMi, fori >- 3, Ai = ([A] +i- 2)-2. Then. `02" p.g A,.,1 pa(9o n Et). A,i2i io-1. s{ pa(9o) log(A) + Ilog([A] + 1)-21 u(E2) + 2 ilog([A] + i - 1)-21 pa(Ei).. i--3.

(10) 10 T. UNo and H. NEGisHi While pa(E2) s (i)ii2 (s;) A-i. and for3si<- io -1 n pa(Ei) S pt({co; .ll.], a.(co)g. s ([A] +i- 2)-2 o}) s (-;;)ii2 ("illi) ([A] + i - 2)'2'. Therefore, we have i,ll.Il,i 1iog Ai.il pt(s2o n E,) s pa(s;2,) iog A + Ce l<og A,. Ai+1<1 where e = (2/z)'i2 (bloh). Hence J., iog .￡., av(co)gv dco }i pa(g,) iog o- pa(g,) iog A - Ce ltog A,. and taking A = 11pt(9o) we see J., iog .￡.oav(oo)gv dco > pa(9o) iog o - ccpt(go) iog (pt(ts,)).. This completes the proof of the lemma. Let N.(r,co) denotes the number of zeros of jE,(z,co) in the circle with center x. and radius r. By Jensen's theorem Jl;II"" 5Nn(,r'to) dr = ". Ji,-.,.i-s,,jz,,..,E, iOg(z,co) E,(x.,co) dZ'. Therefore writing ip.(co) for N.(112M'i,(D), we have ipm((D) S{ 2nlolg(s/4)lz-.,.i=s,,ik,,.,10g ,p"F;Yi,(fZ.'llill) dz. and Jg, ipm((D)dco si 2,t ioig(si4) Jil" de (Jl,, iog ]Fh(x. + 2.5.3 eie, cx)) da). - Jb, ioglEz(xm,co)l dco]. By Lemma 2.2 and 2.3, if S2o n G. = ip, with pa(G.) f{ C. E, where. (Sec-11V-38)-9 (P.11-12). J.

(11) On the Upper Bound of the Number of Positive Roots. 11. e- == (;)if2 (31/L.: .Xll, ii31{fil'i ;:fif':)'i2,. we get J., ￠m(co)dco < 2,,:gSes)14) Jiia log v(x.,e) de + ce.pt(s;2,) log (u(5,) )･. where since O < 6 E{ plf-kl S{ 1 2Joo･,..o x. + 2.5.3 eie ij' + 2 2Joo･<k x. + 2.5.3 eie i"k pif-ki. V(x.,e). 2ioo･=o xZ.i + 2 2,oo.k x'."k pif-kl. (Eioo'=o (xm + 2.5.3)f]2. s 6 (2,oo-=o xh)2 (1 - (1 m 2-1.)]2 6(1 nt (1 m i. '+ 2.5.3)]2. -s (x')･ Hence we obtain f., ipm((D)dcD < Ce.pa(s2o) iog (pt(l},)),. provided 9o n G. = ip, with pt(G.) f{ e./m2, taking E = m-2.. Consider i= f., i(os,,'icoM,'l ,ipMifX, dcD,. where M(co) E {mo, mi, ･･･, M} and. log n - log log log n. M(co)E!irp(n)== 1.g2 ' Ek(co) = {co E 9o; M(co) = k}.. Then. (2.11).

(12) 12 T. UNo and H. NEGisHi tp(n). S2o= U Ek k==mo. and i = il.t(n.', J., 211ZE'{li3,ipM,((D) du) - zi + x2, say,. where Zi contains the terms for which. pa(9o) pa(Ek)S k2 ･ First consider Zi. The function x log x-i is increasing with x for O < x < e-i, and therefore if. pa(k9,O)<e-' or m3>epa(9o), then. u(Ek)iog(pt(},))s2pa(go)iOkg2k+i,pa(go)iog(pa(5,)). (2.i2). '. Now by (2.11) and (2.12), Jl., 2:k-'i 6' gipMk((") d(D = k lolg k .S.l., f., `1'm((D) d`". < k 1.Cg k Pkpa(Ek) 10g (pa(},)). ` C [-ii/÷ -pa-(9o) + k2 f.kg k pa(9o) log (pt(5,))],. if. k. EknH,=￠ with pa(H,)<pk2m-2, m == Mo. where. k. Hk=UG. and Pk=max. e.･. m=mo mosmsk. Now consider ￡2, where pa(Ek) > pa(9o)/k2. If Ek n Hk = ￠, then Jl., 2Xi'{fs,9bMk(a') dco = k i.i, k .S.l., f., ipm(cD) d(o. < 1.gk pkpt(Ek) log (pt(l;,)).

(13) On the Upper Bound of the Number of Positive Roots. 13. s c (pkpt(Ek) + i.Pgkk pa(Ek) iog (pa(l2,)). Hence i :f{ C (,(m.,,.;.k2,--(o.(,y),, (-lil/+ i`(S20) + k2 ;okg k iL(S2o) iog (L,(li,). )). + .(M.o,-;.k2.-<(op.(,y,),, (Pkpt(Ek) + loPgkk pa(Ek) log (pa(l},)))]. provided go n H = ip, where H= U$S-in, G. and. pa(H)gmo-i max e.. Now, since O < 6 s pti-jl S 1,. moSmsO(n). , - 2 2i=, xtt + 2 2,oo･.j xS'ii pl,-fl em - -ii7 2;=, .tt + 2 E)l.f xin'f Pli-fl. 2(2;･2.,,oxjll`)2 2 < -]f 6(2i{=o x:")2 - z6(1 - xth'i)2' Therefore. pk s{ -)i/2 (z26 {1 - (1 - 2-o(n))n+i}-i s (-ii3)'/2 {i - aog no)-'}-'. ' <(-]h-)if2e2 ifno>exp(2-ii2). Hence we have. i < cps(go) iog (pa(5,)) provided 9o n H = ip, with. pa(H) s (e21mo)(2/z6)ii2) = Clmo. Thus we obtain that for n > no and any S2o Jl,, i(MM,,icoM,'s' ,￠M.(C(Dii) d(D < cpt(s2,) iog (,(5,)),. '. if M(co) s{ op(n) and 9o nH = ip, such that pa(H) s C/mo.. ]..

(14) 14. T. UNo and H. NEGisHi Now let 2.M-Scom', ￠m(co) G(co) = .,,,MS(U.?s￠(.) M(co) log M(co) Choose M(co) such that i,-M.icom,'s･ ,￠m>fX, >gG(co). and take S2o = {co; G(co) > ev}. Then we have. evpa(go) < cpa(go) iog (pt(5,)). i.e.. pt(9o) < exp(-drC), outside a set of measure at most Clmo. Thus outside a set of measure at most. C/mo + exp(-cr/C), we get. G(co)sev for n>no. That is. 2m"-Scom', (Pm((D) m,s{MS(Uco?s￠(n) M(co) log M(co) S ev. for n > no.. Hence for all M(co) s O(n). M(co) 2 ip.(co) si evM(co) log M(co) m=:Mo i.e.. M(co). 2 4).((D) E{ ev(log log n) log n, m == Mo outside a set of measure at most. Clmo + exp(-cr/C). Taking cr == log log no, we obtain for all n > no. M(co) 2 (P.(a)) < (log log n)2 log n. m=Mo outside a set of measure at most.

(15) On the Upper Bound of the Number of Positive Roots 15. C log no - log log log no step 5. In this step, we obtain an upper estimate of the number of zeros of E,(z,co) in. Cl. Now IEi(1･(D)1 = 2n a.(co) v =O and. -2exp (-2Uiii) du p({oo; .￡.o a.(co) < (n + i)-2})= (;)ii2 il Ln"). < (i)'/2 (. +11)si2 (2'13) where oZ = (n + 1) +2 ]E) n pli-jr > (n + 1)･. i<1' If Ni denotes the number of zeros of F.(z,co) in Ci, then by (2.4), (2.6) and (2.13), we have for all n > no. 4 log(n + 1) + 2 log Iog n. Nl <. log 2. outside a set of measure at most ...#,.i (-l})i (exp (m(n +2 1)2) + (n +11)si2] < nfti2'. step 6 Now we consider the whole interval [O,1]. If N(co) denotes the number of zeros of E,(z,co)･in [O,1], we have for all n > no outside an exceptional set,. M. N(co)<7Vb+N6+ 2 Nm+Ni m=Mo < (log log n)2 log n. The exceptional set has measure at most i:i + nf/2 (i IOgagOggnnoO)-2)ii2 + iog no - ioCg iog iog no + nft2. <c.. Iog no log ' log log no. Thus the proof of Theorem 2.1 is completed..

(16) 16. T. UNo and H. NEGisHi References. [1] BHARucHA-REiD and SAMBANDHAM, Random Polynomials, academic press, (1986). [2] EvANs, E.A., On the number of real roots of a random algebraic equation, Proc. London Math. Soc. 15, 731-749 (1965). [3] SAMBANDHAM, M., On the upper bound of the number of real zeros of a random aigeb raic eguation. J. Indian Math. Soc. 42, 15-26 (1978).. .. v. i. !. I. i !. l i. 1. e i.

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